Cycles of the second kind for uncertain pendulum-like systems with several nonlinearities
type: Item
instance of: scholarly article
title: Cycles of the second kind for uncertain pendulum-like systems with several nonlinearities
zbMATH Open document ID: 1154.34019
author: Pingli Lu
author: Ying Yang
author: Lin Huang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-06-18T00:00:00Z
review text: The authors study the pendulum-like system with several nonlinearities \[ \begin{aligned}\dot{\sigma}&=\xi,\\ \dot{\xi}&=-\varphi(\xi,\sigma)+x^T C f(\sigma)+g(\sigma),\\ \dot{x}&=Ax+Bf(\sigma)\xi,\end{aligned} \] where \(A\in \mathbb R^{n\times n}\), \(B,C\in \mathbb R^{n\times m}\) are constant matrices, \(f\colon \mathbb R\rightarrow \mathbb R^m\) and \(g\colon \mathbb R\rightarrow \mathbb R\) are \(\Delta\)-periodic functions, and \(\varphi\colon \mathbb R\times \mathbb R\rightarrow \mathbb R\) is a function satisfying \[ \begin{aligned}&\varphi(\xi,\sigma+\Delta)=\varphi(\xi,\sigma)\\ &\mu_1\xi^2\leq \varphi(\xi,\sigma)\xi\leq \mu_2 \xi^2\end{aligned} \] for all \((\xi,\sigma)\in \mathbb R\times \mathbb R\) with constants \(\mu_1,\mu_2\). Using the Kalman-Yakubovich-Popov lemma, the authors derive sufficient conditions for the existence of cycles of the second kind for the above system in the form of matrix inequalities. These results are applied to robust analysis and synthesis of uncertain pendulum-like systems with norm-bounded parameter uncertainties. In particular, it is shown that the static state feedback controller and dynamic output feedback controller problems for a system with uncertainties are equivalent to the same problems for a system without uncertainties.
reviewed by: Peter Giesl
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93B52
zbMATH DE Number: 5290572
zbMATH Keywords: pendulum-like systems
zbMATH Keywords: cycle of the second kind
zbMATH Keywords: analysis and synthesis
zbMATH Keywords: LMI
describes a project that uses: LMI toolbox
MaRDI profile type: Publication
full work available at URL: j.na.2007.04.042
OpenAlex ID: W2065903326
cites work: Frequency-Domain Methods for Nonlinear Analysis: Theory and Applications
cites work: Q4867828
cites work: Q4693183
cites work: Q4830061
cites work: \(H_{\infty }\) controller synthesis for pendulum-like systems
cites work: Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria
cites work: Robust gradient-like property and controller design for uncertain pendulum-like systems
cites work: On the Kalman-Yakubovich-Popov lemma
cites work: Semidefinite programming duality and linear time-invariant systems
DOI: 10.1016/J.NA.2007.04.042
description: scientific article; zbMATH DE number 5290572